Physics 207: Lecture 2 Notes
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Transcript Physics 207: Lecture 2 Notes
Lecture 16
Goals:
• Chapter 12
Extend the particle model to
rigid-bodies
Understand the equilibrium of
an extended object.
Analyze rolling motion
Understand rotation about a
fixed axis.
Employ “conservation of
angular momentum” concept
Assignment:
HW7 due March 25th
After Spring Break Tuesday:
Catch up
Physics 207: Lecture 16, Pg 1
Rotational Dynamics: A child’s toy, a physics
playground or a student’s nightmare
A merry-go-round is spinning and we run and jump on
it. What does it do?
What principles would apply?
We are standing on the rim and our “friends” spin it
faster. What happens to us?
We are standing on the rim a walk towards the center.
Does anything change?
Physics 207: Lecture 16, Pg 2
Rotational Variables
Rotation about a fixed axis:
Consider a disk rotating about
an axis through its center:
How do we describe the motion:
d 2
(rad/s) vTangential /R
dt T
(Analogous to the linear case v
dx )
dt
Physics 207: Lecture 16, Pg 3
Rotational Variables...
Recall: At a point a distance R away from the axis of
rotation, the tangential motion:
v=R
x= R
v=R
a=R
constant
0 t
R
x
2
(angular accelation in rad/s )
(angular v elocity in rad/s)
1
0 0 t t 2 (angular position in rad)
2
Physics 207: Lecture 16, Pg 4
Comparison to 1-D kinematics
Angular
Linear
constant
a constant
0 t
0 0 t t
1
2
v v0 at
2
x x0 v0 t 12 at 2
And for a point at a distance R from the rotation axis:
x=R
v=R
aT = R
Here aT refers to tangential acceleration
Physics 207: Lecture 16, Pg 5
System of Particles (Distributed Mass):
Until now, we have considered the behavior of very simple
systems (one or two masses).
But real objects have distributed mass !
For example, consider a simple rotating disk and 2 equal
mass m plugs at distances r and 2r.
1
2
Compare the velocities and kinetic energies at these two
points.
Physics 207: Lecture 16, Pg 9
System of Particles (Distributed Mass):
1 K= ½ m v2 = ½ m ( r)2
2 K= ½ m (2v)2 = ½ m ( 2r)2
Twice the radius, four times the kinetic energy
K mv m(r )
1
2
2
1
2
2
The rotation axis matters too!
Physics 207: Lecture 16, Pg 10
A special point for rotation
System of Particles: Center of Mass (CM)
If an object is not held then it will rotate about the
center of mass.
Center of mass: Where the system is balanced !
Building a mobile is an exercise in finding
centers of mass.
m1
+
m2
m1
+
m2
mobile
Physics 207: Lecture 16, Pg 11
System of Particles: Center of Mass
How do we describe the “position” of a system made up of
many parts ?
Define the Center of Mass (average position):
For a collection of N individual point like particles whose
masses and positions we know:
mi ri
N
i 1
RCM
M
RCM
m2
m1
r1
r2
y
x
(In this case, N = 2)
Physics 207: Lecture 16, Pg 12
Sample calculation:
Consider the following mass distribution:
N
mi ri
RCM i 1
M
XCM ˆi YCM ˆj ZCM kˆ
XCM = (m x 0 + 2m x 12 + m x 24 )/4m meters
RCM = (12,6)
YCM = (m x 0 + 2m x 12 + m x 0 )/4m meters
(12,12)
2m
XCM = 12 meters
YCM = 6 meters
m
(0,0)
m
(24,0)
Physics 207: Lecture 16, Pg 13
System of Particles: Center of Mass
For a continuous solid, convert sums to an integral.
dm
y
r dm r dm
RCM
M
dm
r
x
where dm is an infinitesimal
mass element.
Physics 207: Lecture 16, Pg 14
Connection with motion...
So for a rigid object which rotates about its center of
mass and whose CM is moving:
K TOTAL K Rotation K Translatio n
K TOTAL K Rotation MV
2
CM
1
2
For a point p rotating:
mi ri
N
RCM i 1
M
K R m p v p m p (rp )
1
2
p
2
1
2
VCM
Physics 207: Lecture 16, Pg 15
2
Rotation & Kinetic Energy
Consider the simple rotating system shown below.
(Assume the masses are attached to the rotation axis by
massless rigid rods).
The kinetic energy of this system will be the sum of the
kinetic energy of each piece:
4
K mi v
1
2
i 1
2
i
K = ½ m1v12 + ½ m2v22 + ½ m3v32 + ½ m4v42
m4
r4
m3
r3
r1
m1
r2
m2
Physics 207: Lecture 16, Pg 16
Rotation & Kinetic Energy
Notice that v1 = r1 , v2 = r2 , v3 = r3 , v4 = r4
So we can rewrite the summation:
4
4
K mi v mi r
1
2
2
i
i 1
1
2
2 2
i
i 1
1
2
4
[ m r ]
i 1
2
i i
2
We recognize the quantity, moment of inertia or I, and
write:
m4
K Rotational I
1
2
2
m3
N
I mi ri
2
r4
r3
r1
r2
m1
m2
i 1
Physics 207: Lecture 16, Pg 17
Calculating Moment of Inertia
N
I mi ri
2
where r is the distance from
the mass to the axis of rotation.
i 1
Example: Calculate the moment of inertia of four point masses
(m) on the corners of a square whose sides have length L,
about a perpendicular axis through the center of the square:
m
m
m
m
L
Physics 207: Lecture 16, Pg 18
Calculating Moment of Inertia...
For a single object, I depends on the rotation axis!
Example: I1 = 4 m R2 = 4 m (21/2 L / 2)2
I1 = 2mL2
m
m
m
m
I2 = mL2
I = 2mL2
L
Physics 207: Lecture 16, Pg 19
Moments of Inertia
For a continuous solid object we have to add up the mr2
contribution for every infinitesimal mass element dm.
An integral is required to find I :
dm
I r dm
2
r
Some examples of I for solid objects:
dr
r
L
R
Solid disk or cylinder of mass M
and radius R, about
perpendicular axis through its
center.
I = ½ M R2
Use the table…
Physics 207: Lecture 16, Pg 22
Exercise Rotational Kinetic Energy
We have two balls of the same mass. Ball 1 is attached
to a 0.1 m long rope. It spins around at 2 revolutions per
second. Ball 2 is on a 0.2 m long rope. It spins around at
2 revolutions per second.
2
K 12 I
What is the ratio of the kinetic energy
of Ball 2 to that of Ball 1 ?
A. ¼
B. ½
C. 1
D. 2
Ball 1
E. 4
I mi ri
2
i
Ball 2
Physics 207: Lecture 16, Pg 24
Exercise Rotational Kinetic Energy
K2/K1 = ½ m r22 / ½ m r12 = 0.22 / 0.12 = 4
What is the ratio of the kinetic energy of Ball 2 to
that of Ball 1 ?
(A) 1/4 (B) 1/2
Ball 1
(C) 1
(D) 2
(E) 4
Ball 2
Physics 207: Lecture 16, Pg 25
Exercise
Work & Energy
Strings are wrapped around the circumference of two solid disks
and pulled with identical forces, F, for the same linear distance,
d.
Disk 1 has a bigger radius, but both are identical material (i.e.
their density r = M / V is the same). Both disks rotate freely
around axes though their centers, and start at rest.
Which disk has the biggest angular velocity after the drop?
W F d = ½ I 2
2
1
(A) Disk 1
(B) Disk 2
(C) Same
start
finish
F
F
d
Physics 207: Lecture 16, Pg 28
Exercise
Work & Energy
Strings are wrapped around the circumference of two solid
disks and pulled with identical forces for the same linear
distance.
Disk 1 has a bigger radius, but both are identical material (i.e.
their density r = M/V is the same). Both disks rotate freely
around axes though their centers, and start at rest.
Which disk has the biggest angular velocity after the drop?
W = F d = ½ I1 12 = ½ I2 22
2
1
1 = (I2 / I1)½ 2 and I2 < I1
(A) Disk 1
(B) Disk 2
(C) Same
start
finish
F
F
d
Physics 207: Lecture 16, Pg 29
Lecture 16
K TOTAL K Rotational K Translatio nal
K TOTAL K Rotational MV
2
CM
1
2
K Rotational I
1
2
Assignment:
HW7 due March 25th
For the next Tuesday:
Catch up
2
I mi ri
2
i
Physics 207: Lecture 16, Pg 30
Lecture 16
Assignment:
HW7 due March 25th
After Spring Break Tuesday: Catch up
Physics 207: Lecture 16, Pg 31